 L11a513 |
 L11a515 |
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The 3-Component Link L11a514Visit L11a514's page at Knotilus! |
 DrawMorseLink |
| PD Presentation: | X8192 X16,5,17,6 X14,3,15,4 X20,12,21,11 X18,10,19,9 X4,15,5,16 X22,18,13,17 X12,20,7,19 X10,22,11,21 X2738 X6,13,1,14 |
| Gauss Code: | {{1, -10, 3, -6, 2, -11}, {10, -1, 5, -9, 4, -8}, {11, -3, 6, -2, 7, -5, 8, -4, 9, -7}} |
| Jones Polynomial: | q-7 - 3q-6 + 7q-5 - 10q-4 + 14q-3 - 15q-2 + 15q-1 - 12 + 10q - 5q2 + 3q3 - q4 |
| A2 (sl(3)) Invariant: | q-22 - q-18 + 3q-16 + 2q-12 + 4q-10 + 5q-6 + q-4 + 4q-2 + 4 + 4q4 + q10 - q12 |
| HOMFLY-PT Polynomial: | - 2a-2z2 - a-2z4 + z-2 + 4 + 4z2 + 3z4 + z6 - 2a2z-2 - 6a2 - 4a2z2 + a2z4 + a2z6 + a4z-2 + a4 - 3a4z2 - 2a4z4 + a6 + a6z2 |
| Kauffman Polynomial: | 4a-3z3 - 4a-3z5 + a-3z7 - 8a-2z2 + 18a-2z4 - 13a-2z6 + 3a-2z8 + 2a-1z3 + 5a-1z5 - 9a-1z7 + 3a-1z9 - z-2 + 7 - 20z2 + 33z4 - 26z6 + 5z8 + z10 + 2az-1 - 9az + 14az3 - 5az5 - 11az7 + 6az9 - 2a2z-2 + 11a2 - 21a2z2 + 23a2z4 - 24a2z6 + 8a2z8 + a2z10 + 2a3z-1 - 9a3z + 22a3z3 - 25a3z5 + 6a3z7 + 3a3z9 - a4z-2 + 3a4 - a4z2 - a4z4 - 5a4z6 + 6a4z8 + 4a5z3 - 8a5z5 + 7a5z7 - 2a6 + 7a6z2 - 8a6z4 + 6a6z6 - 2a7z3 + 3a7z5 - a8z2 + a8z4 |
| Khovanov Homology: |
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Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):
In[1]:= |
<< KnotTheory` |
Loading KnotTheory` (version of August 30, 2005, 10:15:35)... | |
In[2]:= | Length[Skeleton[L]] |
Out[2]= | 3 |
In[3]:= | Show[DrawMorseLink[Link[11, Alternating, 514]]] |
| |
Out[3]= | -Graphics- |
In[4]:= | PD[L = Link[11, Alternating, 514]] |
Out[4]= | PD[X[8, 1, 9, 2], X[16, 5, 17, 6], X[14, 3, 15, 4], X[20, 12, 21, 11], > X[18, 10, 19, 9], X[4, 15, 5, 16], X[22, 18, 13, 17], X[12, 20, 7, 19], > X[10, 22, 11, 21], X[2, 7, 3, 8], X[6, 13, 1, 14]] |
In[5]:= | GaussCode[L] |
Out[5]= | GaussCode[{1, -10, 3, -6, 2, -11}, {10, -1, 5, -9, 4, -8},
> {11, -3, 6, -2, 7, -5, 8, -4, 9, -7}] |
In[6]:= | Jones[L][q] |
Out[6]= | -7 3 7 10 14 15 15 2 3 4
-12 + q - -- + -- - -- + -- - -- + -- + 10 q - 5 q + 3 q - q
6 5 4 3 2 q
q q q q q |
In[7]:= | A2Invariant[L][q] |
Out[7]= | -22 -18 3 2 4 5 -4 4 4 10 12
4 + q - q + --- + --- + --- + -- + q + -- + 4 q + q - q
16 12 10 6 2
q q q q q |
In[8]:= | HOMFLYPT[Link[11, Alternating, 514]][a, z] |
Out[8]= | 2 4 2
2 4 6 -2 2 a a 2 2 z 2 2 4 2
4 - 6 a + a + a + z - ---- + -- + 4 z - ---- - 4 a z - 3 a z +
2 2 2
z z a
4
6 2 4 z 2 4 4 4 6 2 6
> a z + 3 z - -- + a z - 2 a z + z + a z
2
a |
In[9]:= | Kauffman[Link[11, Alternating, 514]][a, z] |
Out[9]= | 2 4 3
2 4 6 -2 2 a a 2 a 2 a 3
7 + 11 a + 3 a - 2 a - z - ---- - -- + --- + ---- - 9 a z - 9 a z -
2 2 z z
z z
2 3 3
2 8 z 2 2 4 2 6 2 8 2 4 z 2 z 3
> 20 z - ---- - 21 a z - a z + 7 a z - a z + ---- + ---- + 14 a z +
2 3 a
a a
4
3 3 5 3 7 3 4 18 z 2 4 4 4 6 4
> 22 a z + 4 a z - 2 a z + 33 z + ----- + 23 a z - a z - 8 a z +
2
a
5 5
8 4 4 z 5 z 5 3 5 5 5 7 5 6
> a z - ---- + ---- - 5 a z - 25 a z - 8 a z + 3 a z - 26 z -
3 a
a
6 7 7
13 z 2 6 4 6 6 6 z 9 z 7 3 7
> ----- - 24 a z - 5 a z + 6 a z + -- - ---- - 11 a z + 6 a z +
2 3 a
a a
8 9
5 7 8 3 z 2 8 4 8 3 z 9 3 9 10
> 7 a z + 5 z + ---- + 8 a z + 6 a z + ---- + 6 a z + 3 a z + z +
2 a
a
2 10
> a z |
In[10]:= | Kh[L][q, t] |
Out[10]= | 8 9 1 1 3 4 3 6 4 8
-- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- +
3 q 15 6 13 6 13 5 11 4 9 4 9 3 7 3 7 2
q q t q t q t q t q t q t q t q t
6 7 8 6 t 2 3 2 3 3 5 3
> ----- + ---- + ---- + --- + 6 q t + 4 q t + 7 q t + 2 q t + 3 q t +
5 2 5 3 q
q t q t q t
5 4 7 4 9 5
> q t + 2 q t + q t |
| Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a514 |
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